If (x-a), (x-b), and (x-c) are factors of a cubic polynomial, which is a possible polynomial?
Answer and explanation
Correct answer: x^3-(a+b+c)x^2+(ab+bc+ca)x-abc
The governing concept is the factor theorem together with expansion of three linear factors. A cubic having factors (x-a), (x-b), and (x-c) can be written as (x-a)(x-b)(x-c), apart from a non-zero constant multiplier. First, (x-a)(x-b) = x^2-(a+b)x+ab. Multiplying by (x-c) gives x^3-(a+b+c)x^2+(ab+bc+ca)x-abc. Thus option A has exactly the required factors and is a possible monic cubic. Option B has incorrect signs, while options C and D place the symmetric sums with incorrect powers or signs. Substituting x=a, b, or c into option A gives zero, confirming each factor.
Frequently asked questions
What is the correct answer to this question?
x^3-(a+b+c)x^2+(ab+bc+ca)x-abc
Why is this the correct answer?
The governing concept is the factor theorem together with expansion of three linear factors. A cubic having factors (x-a), (x-b), and (x-c) can be written as (x-a)(x-b)(x-c), apart from a non-zero constant multiplier. First, (x-a)(x-b) = x^2-(a+b)x+ab. Multiplying by (x-c) gives x^3-(a+b+c)x^2+(ab+bc+ca)x-abc. Thus option A has exactly the required factors and is a possible monic cubic. Option B has incorrect signs, while options C and D place the symmetric sums with incorrect powers or signs. Substituting x=a, b, or c into option A gives zero, confirming each factor.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Polynomials in one variable.
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